Question

Use the following expression to answer the questions below:

(1/x+3 + 6/x^2+4x+3)*x+3/x+1

a. Simplify the expression down to one rational expression.
b. Explain the steps you used to do the simplification.
c. Are there any values x cannot be? Explain.
d. Find the values of x that make the expression equal to 0. Explain the steps you used to find these values.

Answer 1

We want to simplify:


`((1)/(x+3)+ (6)/( x^(2) +4x+3))* (x+3)/(x+1)`

first we factorize
`x^(2) +4x+3` as
`(x+3)(x+1)`

then multiply
`(1)/(x+3)` by
`(x+1)/(x+1)` to equalize the denominators of the first expression, and add the fractions.


`((1)/(x+3)*(x+1)/(x+1) + (6)/((x+3)(x+1)))* (x+3)/(x+1)`

`(x+7)/((x+3)(x+1))* (x+3)/(x+1)`

simplify x+3:


`(x+7)/((x+1))* (1)/(x+1)= (x+7)/( x^(2) +2x+1)`

a.

`(x+7)/( x^(2) +2x+1)`

b.
steps already described

c.
x cannot be -1, nor -3 because the fractions with denominators (x+1) and (x+3) would not be defined.